# Motion in 2 and 3 Dimensions – formula & numerical

Last updated on February 8th, 2024 at 10:40 am

We will discuss and list down the formulae of position, velocity, and acceleration related to the *Motion in two and three Dimensions* using unit vectors i, j, and ** k** format.

We will list down the 2-Dimensional equivalent of motion equations or

*suvat equations*. Also, we will solve numerical problems using the formulas of the position vector, velocity vector, and acceleration vector in 2 Dimension and 3 Dimension.

## Position formula (in 3 dimensions)

In three dimensions, the location of a particle is specified by its **location vector**, **r**:**r** = x **i** + y **j **+ z **k**

If during a time interval ∆t the position vector of the particle changes from r1 to r2, the displacement ∆r for that time interval is represented as: ∆**r** = **r1 − r2 **

= (x2 − x1)**i** + (y2 − y1)**j **+ (z2 − z1)**k**

∆**r**= (∆x)** i **+ (∆y)** j** + (∆z) **k**

## Velocity formulae (in 3 dimensions)

**Average velocity** formula in 3 dimensions

If a particle moves through a displacement ∆**r** in a time interval ∆t then its** average velocity **for that interval is:

** v** =∆

**r**/∆t =(∆x/∆t)

**i**+ (∆y/∆t)

**j**+ (∆z/∆t)

**k**

**Instantaneous velocity** formula in 3 dimensions

A more interesting quantity is the instantaneous velocity **v**, which is the limit of the average velocity when we shrink the time interval ∆t to zero.

**Instantaneous velocity** is the time derivative of the position vector **r**: **v** =d**r**/dt = d(x**i** + y**j** + z**k**)/dt= (dx/dt) **i** +(dy/dt) **j** +(dz/dt) **k**

This **Instantaneous velocity** can be written as:**v** = *v*_{x}**i** +** ***v*_{y}**j** + *v*_{z}**k**

where *v*_{x} = (dx/dt) *v*_{y} =(dy/dt) *v*_{z} =(dz/dt)

## Acceleration formulae (in 3 dimensions)

**Average acceleration** formula in 3 dimensions

If a particle’s velocity changes by ∆v in a time period ∆t, the average acceleration **a** for that period is** a** =∆

**v**/∆t = ∆(

*v*

_{x}

**i**+

*v*

_{y}

**j**+

*v*

_{z}

**k**)/∆t = (∆

*v*

_{x}/∆t)

**i**+(∆

*v*

_{y}/∆t)

**j**+ (∆

*v*

_{z}/∆t)

**k**

**Instantaneous** acceleration formula in 3 dimensions

**Instantaneous**acceleration

But a much more interesting quantity is the result of shrinking the period ∆t to zero, which gives us the instantaneous acceleration, a. It is the time derivative of the velocity vector **v**: **a** =d**v**/dt=d(*v*_{x}**i** +*v*_{y}**j** + *v*_{z}**k**)/dt = (d*v*_{x}/dt)**i** +(d*v*_{y}/dt)**j** + (d*v*_{z}/dt)**k**

which can be written: **a** = a_{x}**i** + a_{y}**j** + a_{z}**k**

where

a_{x} =d*v*_{x}/dt=d^{2}x/dt^{2}

a_{y} =d*v*_{y}/dt=d^{2}y/dt^{2}

a_{z} =d*v*_{z}/dt =d^{2}z/dt^{2}

## 2 dimensional Motion equations in vector format with Constant Acceleration

**2-D motion equations** – Let’s talk about the Motion equations in 2 dimensions when we have Constant Acceleration in Two Dimensions.

When the acceleration **a** (for motion in two dimensions) is constant we have two sets of equations to describe the x and y coordinates, each of which is similar to the * suvat equations*.

In the following discussion, the motion of the particle begins at t = 0.

The initial position of the particle is given by **r**_{0} = x_{0}**i** + y_{0}**j**

and its initial velocity is given by **v**_{0} = v_{0x}**i** + v_{0y}**j**

and the acceleration vector **a** = a_{x}**i** + a_{y}**j** is constant.

## Worked Examples | Solved Numerical problems

Let’s solve a few selected numerical problems on *Motion in 2 and 3 Dimensions* using the formulae of the **position vector**, **velocity vector**, and **acceleration vector** in 2 Dimensions.

- The position of an electron is given by
**r**= 3.0t**i**− 4.0t^2**j**+ 2.0**k**(where t is in seconds and the coefficients have the proper units for**r**to be in meters).

(a)What is**v(**t) for the electron? (b) In unit–vector notation, what is**v**at t = 2.0 s?

(c) What are the magnitude and direction of**v**just then?**Solution:**

- A particle moves so that its position as a function of time in SI units is
**r**=**i**+ 4t^2**j**+ t**k**. Write expressions for (a) its velocity and (b) its acceleration as functions of time.

Solution: